This work continues researches in developing models of swarming and aggregation in flows with dispersed systems. In the submitted work the new approach based on the method of random walk on stochastic lattices to computer simulation of attractive swarming, accompanied by particle aggregation, for the case of a moving region of attraction has been applied. Attractors of this type can arise, for example, in flows of charged particles or in biological systems. The competition of various control parameters has been considered. They are: the rate of entrainment of dispersion particles and attractive region by a continuum media flow in the apparatus; random drift of particles, which simulates the influence of Brownian motion and turbulent pulsations; diffusion limited kinetics of particle aggregation with the formation of clusters with account of the influence of moving attractor. The principal novelty of the modified algorithm is that the influence of the moving attractor on the drift of particles is described by the prevalence of the transition probability from a given cell to that lattice cell adjacent to it, which is located closer to the attractor. A justification for the developed algorithm, the results of numerical experiments, and their interpretation have been described. The results of the work can be useful in the design of technological apparatuses for the fine chemical and pharmaceutic technology or energetics.